Multilinear Hardy-Cesàro Operator and Commutator on the product of Morrey-Herz spaces

Abstract

We obtain sufficient and necessary conditions on weight functions $s_1(t),\dots ,s_m(t)$ and $\psi (t)$ so that the weighted multilinear Hardy-Cesàro operator \[(f_1,\ldots,f_m)\mapsto \int_{[0,1]^n}\left(\prod_{k=1}^nf_k\left(s_k(t) x\right)\right)ψ(t)dt \] is bounded from ${\dot{K}}_{q_1}^{{\alpha }_1,p_1}({\omega }_1)\times \cdots \times {\dot{K}}_{q_m}^{{\alpha }_m,p_m}({\omega }_m)$ to ${\dot{K}}_q^{\alpha ,p}(\omega )$ and from $M{\dot{K}}_{p_1,q_1}^{{\alpha }_1,{\lambda }_1}({\omega }_1)\times \cdots \times M{\dot{K}}_{p_m,q_m}^{{\alpha }_m,{\lambda }_m}({\omega }_m)$ to $M{\dot{K}}_{p,q}^{\alpha ,\lambda }(\omega )$. The sharp bounds are also obtained and these results hold for both cases $0<p<1$ and $1\le p<\infty$. We give a sufficient condition so that if symbols $b_1,\dots ,b_m$ are Lipschitz, then the commutator of the weighted Hardy-Cesàro operator \[ (f_1,\ldots,f_m)\mapsto\int_{[0,1]^n}\left(\prod\limits_{k=1}^mf_k\left(s_k(t)x\right)\right)\left(\prod_{k=1}^m\left(b_k(x)-b_k\left(s_k(t)x\right)\right)\right)ψ(t)dt\] is bounded from $M{\dot{K}}_{p_1,q_1}^{{\alpha }_1,{\lambda }_1}({\omega }_1)\times \cdots \times M{\dot{K}}_{p_m,q_m}^{{\alpha }_m,{\lambda }_m}({\omega }_m)$ to $M{\dot{K}}_{p,q}^{{\alpha }^{\prime },\lambda }(\omega )$ for both cases $0<p<1$ and $1\le p<\infty$. By these we extend and strengthen previous results deu to Tang, Xue, and Zhou [16].

References in corpus (1)

• Weighted multilinear Hardy operators and commutators